What kinds of problems can AI solve without using a deep neural network?
I was hoping to see more answers here, but I'll get us started with some examples:

Combinatorial Search Problems: If your problem can be phrased as movement through a combinatorial graph, you don't need a neural network. In particular, your problem should have discrete states, a clear set of actions that are possible in each state, a clear definition of where we start, and a clear definition of what the goal state looks like. The most effective general purpose technique is iterative deepening search. If you have an idea about which moves might be more effective, or better, a function that estimates how far each state is from the goal, you may be able to build a heuristic function and use A* search instead. Common applications for these techniques include pathfinding in video games (or directions in other applications), AI planning, and Automated Theorem Proving.

I'll add some more topics later, but I suspect others have expertise to share here. Let's see some more ideas!